JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 85, 461472 (1982)

Diffusion and Convection in a Family of Tubes

J. B. GARNER

Se&on on Theoretical Bioihysics, National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland 20205, and Louisiana Tech University, Ruston,

Louisiana 71272

R. B. KELLOGG

Section on Theoretical Biophysics, National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland 2020.5, and IPST, University of Maryland, College

Park, Maryland 20742

Submitted by E. Stanley Lee

Global existence theorems are given for the solutions of the renal flow equations for a system of parallel tubes. In the system, a solute moves in a fluid through the tubes by diffusion and convection and the tubes exchange fluid and solute with each other through the tube walls. The proofs use a fixed point argument. It is shown that the hypotheses of the theorems include the passive and convective flux formulas used in renal models.

1. INTRODUCTION

In recent years, there has been an effort to understand the working of the kidney in terms of mathematical models. These models are formulated in terms of the diffusion and convection of water and solutes through a network of tubes, corresponding to the system of nephrons, vasculature, and interstitial space of the kidney. The tubes exchange fluid and solute with each other through the tube walls. The phenomenological laws governing this transmembrane water and solute exchange are of vital importance for the model. The concentrations and flows in the tubes are determined by means of a system of differential equations, the renal flow equations. We are concerned with a mathematical analysis of this system of equations.

In [2,3], we have considered flow of water and a solute in a single tube

* Work supported in part by N.I.H. Grant 1 ROl AM20373.

461

409/85/2-12

0022-247X/82/020461-12$02.00/0 Copyright 0 1982 by Academic Press, Inc.

All rights of reproduction in any form reserved.

462 GARNER AND KELLOGG

that exchanges water and solute through the tube wali with a bathing interstitium of known solute concentration. Existence and uniqueness theorems are obtained for the solution of the problem, under certain hypotheses on the transmembrane flux formulas. In [l] there is given a local existence and uniqueness theorem for flow in a system of n parallel flow tubes. In this work, the effect of solute diffusion is neglected. A lower bound is required for the inlet flows, and the implicit function theorem is used in the proof.

In this paper we also consider flow in a system of n parallel tubes. With certain conditions on the transmembrane flux formulas, we give a global existence theorem for the solution of the renal flow equations. The proof of our result uses a fixed point argument. We discuss the extent to which our conditions are satisfied by the flux formulas used in renal models. Basically, our conditions are designed to handle the case of “passive transport,” in which the solute and water transport across a membrane are driven by the difference in solute concentrations on either side of the membrane. In this case, we show that the system does not concentrate, in the sense that the solute concentration at any point is not larger than the largest specified inlet concentration. This generalizes a result of Stephenson [6]. If active solute transport across a membrane is allowed, the existence theory is still not resolved. We consider both the case of positive solute diffusion and zero solute diffusion. In the latter case, we also require that the inlet flows be sufficiently large. The effect of this requirement is to rule out the possibility of a stagnant point, where one of the flows vanishes.

In Section 2 we formulate the system of equations under consideration. Sections 3 and 4 give the results for position diffusion and zero diffusion, respectively, and Section 5 gives a result that allows both possibilities. In Section 6 we give a discussion of the flux formulas.

2. THE RENAL FLOW EQUATIONS

We study a bundle of n, roughly parallel flow tubes, in each of which a water-solute mixture is flowing. A pair of tubes may have a common boundary along their length; if so, the tubes may exchange water and solute through the common wall. In this case, we say that the tubes are linked. This linkage relation defines an undirected graph G on n nodes. If the bundle of tubes were parallel, topological considerations would dictate that a tube could be linked to at most three other tubes. However, anatomical studies show that the renal tubules have a more convoluted structure (or braiding). To represent this, we allow the linking between the tubes in the bundle to be represented by an arbitrary graph G. (See Stephenson [7, p. 5151.) For some good illustrations of the tubule structures in the kidney, see [4].

DIFFUSION AND CONVECTION 463

We let x denote the axial variable along the length of the tube bundle, 0 <x < 1, and we let C,(x) and F,(x) denote respectively the steady-state solute concentration and volume flow in the ith tube. We denote by JSv(x, C,, Cj) and Jou(x, C,, C,), respectively, the transmembrane solute flux and volume flux from tube i to t&j. We set JSii = JUii = 0 and if tubes i and j have no common boundary, we set J,, = J,, = 0. Then

Jsi = f Jsii and J,,i = 2 .I,, j=l j=l

denote respectively the total transmembrane solute flux and volume flux out of tube i. Letting Di > 0 be the constant diffusion coefficient of the solute in tube i, we find the system of differential equations for the C,(x), Fi(x), i = l,..., n, to be

-DiC/ f (FiCi)’ = -Jsi(Xy Cl ye*.) C”), (1)

F; = -Jo*(x, c, )...) C,). (2)

Along with this system of differential equations we impose the boundary conditions as defined in the following description of the system of tubes.

Boundary Conditions

At x = 0 certain pairs of tubes will be considered as attached to each other. The remaining tubes will be either open or closed at x = 0. The same type conditions will hold at x = 1. The collection of tubes may be divided into sequences, such that in each sequence, each tube is attached to the following tube at x = 0 or x = 1. It is assumed that each sequence either starts or ends with a tube with an open end. In particular, a tube with two closed ends is not allowed. If tubes i and j are attached at x = 0 or x = 1, then either Di = Dj = 0 or DiD, > 0 and the boundary conditions there are

ci = cj, DiCf = -DjC;, Fi = -Fj.

If tube i is closed at one end, we require D, > 0. The boundary conditions are

DiCf = 0 and Fi = 0.

If tube i is open at x = 0 or x = 1 and if Di > 0, the value of the concen- tration there is a specified number

C,(O) = Ci, > 0 or C,(l) = ci, > 0.

If Di = 0 and both ends are open, then only one of these conditions is imposed. If Di = 0 and one end of tube i is open, the boundary conditions

464 GARNER AND KELLOGG

are more complicated and are described as follows. By following a tube i from its open end through perhaps a series of attached tubes, attached by pairs at the boundaries x = 0 and x = 1, one reaches a tube, say k, with an open end. The boundary condition is then either a specified value for Fi at the open end of tube i or for Fk at the end of tube k, but not both, along with the specified concentration at one open end.

Our existence result will depend upon certain conditions on the functions K, and Ki where

and

K,(X, Ci, Cj) = Jsii(X, Ci, Cj) - CiJuij(X, Ci, Cj>

n Ki = c K,.

j=l

We will assume there exist constants M, , Mz, with

M, < min{given concentrations at x = 0 and x = 1 },

M, > max(given concentrations at x = 0 and x = 1)

such that:

I-i,. For each i, j, x, 1 < i, j < n, 0 < x < 1, K,(x, M, , Cj) < 0 for M, < Cj < M, . One has equality if and only if Cj = M, .

Hz. For each i, j, x, 1 <i, j<n, O<x< 1, K,(X,M*,Cj)~O for M, < Cj < M, , One has equality if and only if Cj = M, .

In addition, for these constants M, , M, we will always assume that each JSij, JUi, is continuous in (x, Ci, Cj) for 0 < x < 1, 44, Q Ci, Cj Q M, and satisfies a Lipschitz condition with respect to Ci, Cj there.

In renal models, abrupt changes in tubular cross section and wall thickness are often treated by allowing axial discontinuities in the diffusion coefficients Di and the functions Jsij, Jvij. When this is the case, the first term of (1) must be written in conservation form, as -(DiCi)‘. Our arguments can be extended to handle these piecewise continuous situations.

3. POSITIVE DIFFUSION

We now assume that each Di > 0 and consider the problem (1 ‘), (2) and the above described boundary conditions, where

-D, C; + t(F,C,)’ = -tJsi(x, C, ,..., C,), O<t< 1, i= 1,2 ,..., n.(l’)

Our result will require the following two lemmas.

DIFFUSION AND CONVECTION 465

LEMMA 1. The boundary value problem consisting of

cp=o, i = 1, 2,..., n

with the homogeneous boundary conditions determined from the conditions for the C,(x) by replacing all specified concentrations by zero has only the tn’vial solution.

ProoJ Suppose the problem has a nontrivial solution and assume

cjtXd = oFxy, {C,(x), i= 42 ,..., n) > 0.

Then since each C,(x) is a linear function, X, = 0 or X, = 1. Now, x, cannot be an open end of tube j for in this case Cj(x,) = 0. Also, X, cannot be a closed end of tube j for then Cj(x,) = 0 and this and the homogeneous boundary condition yield Cj(x) s 0. Thus x, must be an attached end of tube j with that of, say, tube k, and Cj(Xl) # 0. Then either Cj( 1) > 0, C,(l) < 0 or C,(O) < 0, C,(O) > 0 and we have C,(x) > Cj(x), x # X, . Thus, Cj(x,) = 0. In a similar manner

C,(x,) = ,t$hl (C,(x), i = 1,2 ,..., n}

can be shown to be zero. The lemma follows.

LEMMA 2. Assume H, and H, hold and that for fixed t E [O, 11, tc,tx),..*, C”(X), F,(x),..., F,(x)) is a solution of (1’), (2), and the associated boundary conditions, such that M, < C,(x) <M, on 0 <x < 1 for each i. Then M, < C,(X) < M,.

Proof. If t = 0, then (1’) reduces to Cf’ = 0 and thus C,(x) = a,x + bi, i = 1, 2,..., n. By Lemma 1, the ai, bi are uniquely determined by the boundary conditions. Also since each specified concentration is between M, and M,, we find M, < C,(X) < M, on [0, l] for each i. We now consider the case t # 0.

Assume there exists some p, 1 <p < n, and x,,, 0 <x,, < 1, such that C,(x,) = M,. If 0 < x,, < 1, then

qx,) = 0, Cjyx,) < 0. (3)

If x0 = 0 or x,, = 1, then x,, must be either an attached end or a closed end of tube p. Using the boundary conditions for the attached end or closed end, we again get (3). In terms of Ki we can combine (1’) and (2) to obtain

-DC; + tF, C/ = -tK,, i = 1, 2 ,..., n. (4)

466 GARNER AND KELLOGG

Taking x = x0 in (4) and using (3) we find

From Hi, KP,(xO, M, , C,(x,)) = 0. Hence, again from Hi, Cj(xO) = M, , C,l(xo) = 0, for each j which is linked top. Since the minimum M, is attained in the tube j at x =x6, we may apply this argument again. We ultimately

find that Ci(xo) = M, , C;(x,) = 0 for each i E H = the connected component of node p in the linkage graph G. Upon considering the initial value problems consisting of the equations from (4) for i E H, and the conditions

Ci(xO) = M, 9 c;(x,) = 0, i E H,

we conclude that each C,(x) s M, . This contradicts the boundary conditions if any tube in this subsystem has an open end. If not, tube p is connected, through perhaps a series of attached tubes in this subsystem, to a tube, say tube r, not in the subsystem. B continuity of concentration at connected ends, we have C, =M, at this connected end. We now repeat the above argument for this C,(x). By continuing in this manner we eventually obtain a Ki < 0 or obtain an open end with concentration M,. In either case, a contradiction to C,(x,) = M, is obtained. Thus, each C,(x) > M, on [0, 11. By using H, a similar argument establishes C,(x) < M, on [0, 11.

We are now ready to establish the main result.

THEOREM 1. Assume that H, and H, hold. Then there exists a solution of (I), (2), satisfying the associated boundary conditions.

ProoJ: Let X be the Banach space of continuous functions C(x) = (C,(x),..., C,(x)) on [0, I] with the standard norm. Let Zc X denote the closed convex set defined by M, Q C,(x) < M2. For a given c(x) E 2, define &.g = (F&)9..., I;,(x)) as the corresponding solution of (2) and the associated boundary conditions for the F,(x). It is an easy matter to show that these conditions determine unique constants a, such that

Fi(x) = a, - cx Jui(u, c(u)) du. JO

Now for this c(x), F(x) we define c*(x, t) = U,((?(x)) by the equations

c:(x, t) = 61: [F,(r) c,(r) + j),,(u, ~W> du ] dr + w + bi9

(5)

where, by Lemma 1, the ai, bi are constants uniquely determined by the boundary conditions.

DIFFUSION AND CONVECTION 467

We have now defined a continuous mapping U, of 2 x (0, 1] into a subset of X. It can be shown that U,(Z x [0, 11) is compact. Since Ci(x, 0) = six + b,, it follows from the boundary conditions that

M, < cfyx, 0) < M,, o<x< 1, i = l,..., It,

for all c(x) E Z. Thus, U,(aZ) c Z, where aZ denotes the boundary of Z. Suppose U, has a fixed point on aZ for some t = t, , say c(x, 1,). Then this

function along with the corresponding F(x, t), determined by (5), forms a solution of the problem as in Lemma 2 for t = t,. Thus, for each i, M, < C,(x, tl) < M, and we have a contradiction to our assumption.

The Browder-Potter theorem [5, p. 301 applies to yield a fixed point of U, and, in turn, a solution of the problem.

Remark 1. The preceding proofs remain valid when the boundary conditions at the attached ends are generalized ‘LO

ci = cj, DiC; = -pjDjCj, Fi = -q,F,,

where pi, qj are positive constants.

4. ZERO DIFFUSION

If the flow in a tube is large enough, the convection term dominates the diffusion term and the equation may be somewhat simplified by taking D = 0 in the tube. To treat this, we consider first the system where zero diffusion is taken in each tube, i.e., Di = 0, 1 < i < n. We impose an additional hypothesis that requires the given flow rates at the boundaries to be sufficiently large in magnitude. The effect of this hypothesis is to ensure that the flow never vanishes; i.e., there are no stagnant points in the system. In addition, we assume that tubes with a closed end are not present in the system, and we consider only the case when concentrations are specified at inflowing boundaries.

We state these conditions in the following manner. Let p, q E (0, 1 } with p f q. If any tube, say tube r, is open at x =p with a given flow F, assigned there and if tubes r = r,,, r , ,..., rk are a sequence of attached tubes such that the last tube, tube rk, in the sequence is either open at p or q, then we require

H,. CE z,

There is an a > 0 such that, for i = 0, l,..., k, x E [O, 1 ] and

468 GARNER AND KELLOGG

if i is even and

if i is odd. We require the prescribed concentrations to be such that:

If (-1)” Fr > 0, then C,(p) is prescribed; if (-l)P F, < 0, then k > 0 and C, is prescribed at the open end of tube rk.

Condition (6) assures that only inlet concentrations are specified. For our fixed point argument we consider the system

(6)

F&f = - tKi, t E [O, 11, i = 1, 2 ,..., n.

(7) F,! = - J”i, (8)

LEMMA 3. Assume H,, H,, and H, hold for some M,, M,, where H, applies to each tube with a giuen flow at a boundary. For fixed t E [0, 1 ] let (c(x), F(x)) be a solution of (7), (8) and the associated boundary conditions satisfying (6) such that M, ,< C,(x) (M, on [0, 1). Then M, < C,(x) < Mz.

Proof. Upon considering (8) with the flow boundary conditions, we find that all Ft(x) are different from zero on [0, 11. Thus, for t = 0, (7) reduces to Cl = 0. Thus each C,(x) must equal to one of the inlet concentrations.

For the case of t # 0, assume there exist j and x, such that Cj(xO) = M, . Then, through a consideration of the geometry of the tubes as well as (6), we determine, for Cj(xO) = M, ,

F,(x,) C, (~g) = -tKj(xo ) ~(Xx,) ~ 0

and this contradicts H, or the uniqueness of an initial value problem as in the proof of Lemma 2. The inequalities C,(x) < M, follow by a similar argument.

THEOREM 2. Assume H, , H,, and H, as in Lemma 3. Thn there exists a solution of (I), (2) satisfying the boundary conditions subject to (6).

Proof: The proof is similar to that of Theorem 1 where here the mapping c*(x, t) = V,(@)) is defined by the equations

F,CT’ = -tK,(x, c(x)),

F! = J,i(X, I)

and the boundary conditions for the problem imposed on c*(x, t), g(x).

DIFFUSION AND CONVECTION 469

5. POSITIVE DIFFUSION IN SOME TUBES

AND ZERO DIFFUSION IN REMAINING ONES

The proofs of the preceding two sections can be merged to establish the existence of a solution of the problem as originally described in Section 2. Without further proof we state this result as

THEOREM 3. Assume H, and H, hold for some M,, M,. For these constants, assume H, holds when applied to all tubes with zero dtrusion. Suppose tubes with a closed end have positiue dt@ision and that the boundary conditions for the zero di@sion tubes satisfy (6). Then there exists a solution of (l), (2) satisfying the boundary conditions.

6. FLUX FORMULAS

A typical set of transmembrance flux formulas used in kidney modeling in which our results apply is as follows [ 21:

where

and either

or

Jsij(ci, Cj) = Jp*j(ci 3 Cj) + Jcij(ci 3 cj>,

J,,ij(Ci, Cj) = -o,h,tj(Ct - Cj), i,j= 1,2 ,..., n, (9)

Jpij = h,,(C; - Cj)

Jcij = ~ (1 - ai)(Ci + Cj) Juij (11)

Ci - Cj Jcij = t1 - Oil ln ci _ ln cj Jo,* (11’)

The parameters hst,, h,, are all nonnegative and 0 < u1 < 1. The expressions for Jp,j and Jc,, give contributions to the solute flux arising from passive transport and convective transport, respectively, through the wall of the tubes.

In order to determine if these formulas satisfy H, and H, we compute Kij(x, C,C,). For (9), (lo), ad (11)

K, = (Ci - Cj)(h,ij + joth”tj{( 1 + Ui) Ci - (1 - oi) Cj}) (12)

470 GARNER AND KELLOGG

and for (9), (lo), and (11’)

KU = CC, - C,)(hs, + O,ho,j{Ci - (1 -~i)f(C,v Cj)}),

where f(u, u) = (u - v)/(ln u - In u). We have immediately

(13)

THEOREM 4. In the case of (12), H, is satisfied for any M, and H, is satisfled provided

hsij + aihuij I

1 +a. YM,- qM2 >o,

I i = 1, 2 ,..., n.

For the case of (13) we use the following properties off(u, u):

(i) f(u, u) > 0 for u, zI > 0, (ii) f(u, v) = 0 for uu = 0, (iii) f(u, u) = u for u > 0, (iv) u -f(u, V) > 0 for u > u > 0.

These properties are established by writing f(u, V) = ug(u/v), where

g(l) = l$-’ and studying g(A) for L > 1.

THEOREM 5. In the case of (13) H, is satisfied for M, = 0 and H, is satisfied for any M, .

Proof. From (ii) we conclude that K,(X, M,, Cj) < 0 for M, = 0. From (iv) we have

hsij + uihuij{M2 - ( 1 - ui)f(“*, Cj)} > h,ij t Uih,ij{Mz -f(Mz, Cj) )>O

for any M, > Cj. Moreover, K,(x, M,, Cj) # 0 for Mk # Cj, k = 1,2, and the theorem follows.

Remark 2. Since Juij is a linear function, it is an easy matter, after the geometry of the system of tubes is given, to write inequalities in M, , M, , ui, h vij, and the given flow rates which imply H, ,

Remark 3. Since, in Theorem 4 and Theorem 5, H, is satisfied for an M2 satisfying

M, > maxjgiven concentrations at x = 0 and x = 11,

the strict inequalities M, < C,(x) < M, can be obtained in Lemmas 2 and 3

DIFFUSION AND CONVECTION 471

by assuming only that M, < C,(x) instead of M, < C,(x) GM,. Therefore, we conclude that a system of flow tubes with flux terms given by (9), (10) with either (11) or (11’) cannot concentrate. Moreover, in case of only passive transport (all ui = l), Lemmas 2 and 3 can be obtained for any M, such that

M, < min{ given concentrations at x = 0 and x = 11

by assuming 0 < C,(x) instead of M, < C,(X) GM,. In this case the system cannot dilute or concentrate. This generalizes a result of Stephenson [ 61 where each Di was assumed to be zero and where each Fi(x) was taken as a known constant.

Remark 4. Other typical sets of flux formulas include terms representing solute flux arising from active transport (see [2]). In this case Jsij, as given by (9), contains an additional term of the form

L2.C. JOij(Ci, Cj) = L!E!- - --L-L- bi + Cj bj + Cj

given by Michaelis-Menten kinetics. Further complications that occur in renal models include the presence of more than one solute and the presence of hydrostatic pressure as an additional variable. As yet, our results have not been extended to these cases because of difficulties in obtaining a priori bounds for the solution. In some of these more complicated models, there is numerical evidence to suggest that, for some parameter values, there may not be any solutions with positive concentrations. Also, in some cases one can show that the models admit multiple solutions. These multiple solutions are presently being studied. In these cases, the maximum concentrations may exceed the inlet concentrations; in fact, the study of this concentrating power of the kidney is one of the main goals of renal modeling.

ACKNOWLEDGMENTS

We thank Dr. .I. L. Stephenson for a number of comments on the paper.

REFERENCES

1. P. FARAHZAD, Analysis of the equations of renal network flows, Math. Biosci. 40 (1978), 233-242.

2. J. B. GARNER AND R. B. KELLOGG, A one tube flow problem arising in physiology, Bull. Marh. Biol. 42 (1980), 295-304.

3. J. B. GARNER AND R. B. KELLOGG, The diffusionxonvection equation with pressure, J. Math. Anal. Appl. 79 (1981), 58-70.

472 GARNER AND KELLOGG

4. W. KRIG AND A. F. LEVER, Renal countercurrent mechanisms: Structure and function, Amer. Heart J. 78 (1969), 101-118.

5. D. R. SMART, “Fixed Point Theorems,” Cambridge Univer. Press, Cambridge, 1974. 6. J. L. STEPHENSON, Concentration in renal counterflow systems, Biophys. J. 6 (1966),

539-551. 7. J. L. STEPHENSON, Concentrating engines and the kidney. Biophys. J. 13 (1973), 5 12-545.